Mapping/Squaring/Injective and surjective/Example

The mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is neither injective nor surjective. It is not injective because the different numbers ${}2$ and ${}-2$ are both sent to ${}4$. It is not surjective because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is injective, but not surjective. The injectivity can be seen as follows: If ${}x\neq y$, then one number is larger, say

${}x>y\geq 0\,.$

But then also ${}x^{2}>y^{2}$, and in particular ${}x^{2}\neq y^{2}$. The mapping

$\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is injective and surjective.